Transfer matrices, non-Hermitian Hamiltonians and Resolvents: some spectral identities

TL;DR

This paper derives spectral identities linking transfer matrices, resolvents, and Hamiltonian eigenvalues, providing tools for analyzing spectral properties of non-Hermitian systems in quantum transport and localization.

Contribution

It introduces explicit representations of transfer matrices in terms of resolvent blocks and establishes exact relations between their traces and Hamiltonian spectra, extending to non-Hermitian cases.

Findings

Derived explicit transfer matrix representations using resolvent blocks

Established trace relations between transfer matrices and Hamiltonian eigenvalues

Extended identities to non-Hermitian transfer matrices T^d7 T

Abstract

I consider the N-step transfer matrix T for a general block Hamiltonian, with eigenvalue equation L_n \psi_{n+1} + H_n \psi_n + L_{n-1}^\dagger \psi_{n-1} = E \psi_n where H_n and L_n are matrices, and provide its explicit representation in terms of blocks of the resolvent of the Hamiltonian matrix for the system of length N with boundary conditions \psi_0 =\psi_{N+1} =0. I then introduce the related Hamiltonian for the case \psi_0 = z^{-1} \psi_N and \psi_{N+1} = z \psi_1, and provide an exact relation between the trace of its resolvent and Tr(T-z)^{-1}, together with an identity of Thouless type connecting Tr(\log |T|) with the Hamiltonian eigenvalues for z=e^{i\phi}. The results are then extended to T^\dagger T by showing that it is itself a transfer matrix. Besides their own mathematical interest, the identities should be useful for an analytical approach in the study of spectral properties of a physically relevant class of transfer matrices. P.A.C.S.: 02.10.Sp (theory of matrices), 05.60 (theory of quantum transport), 71.23 (Anderson model), 72.17.Rn (Quantum localization)